Some algebraically compact modules. I.

*(English)*Zbl 0848.16011
Facchini, Alberto (ed.) et al., Abelian groups and modules. Proceedings of the Padova conference, Padova, Italy, June 23-July 1, 1994. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 343, 419-439 (1995).

Let \(A = kQ/I\) be a finite-dimensional monomial algebra over a field \(k\). It is well-known [see M. C. R. Butler and C. M. Ringel, Commun. Algebra 15, 145-179 (1987; Zbl 0612.16013)] how to associate to a finite word in the arrows of \(Q\) and their formal inverses a finite-dimensional indecomposable \(A\)-module – a “string” module. The author extends this association to (doubly or singly) infinite eventually periodic words.

Any such word \(x\) is already periodic or contains a maximal periodic subword \(y\) (possibly empty) on either side. An important distinction is drawn between the case where \(y\) is “expanding” and that where \(y\) is “contracting”. Thus a singly infinite eventually periodic word is either expanding or contracting and a doubly infinite word is expanding (on both sides), contracting or “mixed”.

From any infinite word \(x\) one may define in the obvious way the infinite-dimensional direct sum string module \(M(x)\) and product module \(\overline{M} (x)\). Using these one then associates to the eventually periodic word \(x\) the module \(C(x)\) which is defined to be \(\overline{M} (x)\) if \(x\) is expanding, \(M(x)\) if \(x\) is contracting and a module which is (roughly) on one side direct sum, on the other side direct product if \(x\) is mixed.

The main theorem of the paper is that the module \(C(x)\) is algebraically compact (=pure-injective) and even \(\Sigma\)-algebraically compact if \(x\) is contracting.

To prove this one uses that an \(A\)-module is algebraically compact if it is linearly compact over its endomorphism ring. Specifically, it is shown that the module \(C(x)\) is linearly compact over its “shift ring”. This is a subring of the endomorphism ring of \(C(x)\) which is generated by the ring of formal power series in the “shift” endomorphism(s) of \(C(x)\): a shift endomorphism is defined using the eventual periodicity of \(x\).

The paper also describes the Prüfer, \(p\)-adic and generic infinite-dimensional, algebraically compact modules associated to a primitive cyclic word.

Then the author specialises to string algebras and shows, in particular, that over such an algebra every infinite word is eventually periodic iff there are only finitely many primitive cyclic words (and then the number of equivalence classes of these is at most the number of simple modules).

The paper closes with a collection of examples which give concrete form to the various kinds of words and modules described.

For the entire collection see [Zbl 0830.00031].

Any such word \(x\) is already periodic or contains a maximal periodic subword \(y\) (possibly empty) on either side. An important distinction is drawn between the case where \(y\) is “expanding” and that where \(y\) is “contracting”. Thus a singly infinite eventually periodic word is either expanding or contracting and a doubly infinite word is expanding (on both sides), contracting or “mixed”.

From any infinite word \(x\) one may define in the obvious way the infinite-dimensional direct sum string module \(M(x)\) and product module \(\overline{M} (x)\). Using these one then associates to the eventually periodic word \(x\) the module \(C(x)\) which is defined to be \(\overline{M} (x)\) if \(x\) is expanding, \(M(x)\) if \(x\) is contracting and a module which is (roughly) on one side direct sum, on the other side direct product if \(x\) is mixed.

The main theorem of the paper is that the module \(C(x)\) is algebraically compact (=pure-injective) and even \(\Sigma\)-algebraically compact if \(x\) is contracting.

To prove this one uses that an \(A\)-module is algebraically compact if it is linearly compact over its endomorphism ring. Specifically, it is shown that the module \(C(x)\) is linearly compact over its “shift ring”. This is a subring of the endomorphism ring of \(C(x)\) which is generated by the ring of formal power series in the “shift” endomorphism(s) of \(C(x)\): a shift endomorphism is defined using the eventual periodicity of \(x\).

The paper also describes the Prüfer, \(p\)-adic and generic infinite-dimensional, algebraically compact modules associated to a primitive cyclic word.

Then the author specialises to string algebras and shows, in particular, that over such an algebra every infinite word is eventually periodic iff there are only finitely many primitive cyclic words (and then the number of equivalence classes of these is at most the number of simple modules).

The paper closes with a collection of examples which give concrete form to the various kinds of words and modules described.

For the entire collection see [Zbl 0830.00031].

Reviewer: M.Prest (Manchester)

##### MSC:

16G20 | Representations of quivers and partially ordered sets |

03C60 | Model-theoretic algebra |

16P10 | Finite rings and finite-dimensional associative algebras |

68R15 | Combinatorics on words |

16D50 | Injective modules, self-injective associative rings |